(e129) 
VIII.—An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate. 
By John Dougall, M.A. Communicated by Professor G. A. GIBson. 
(MS. received February 4, 1904, Read March 21, 1904, Issued separately August 5, 1904.) 
The following paper contains a purely analytical discussion of the problem of the 
deformation of an isotropic elastic plate under given forces. The problem is an unusually 
interesting one. It was the first to be attacked (by Lamé and CLAPEYRON in 1828) after 
the establishment of the general equations by Navimr. The solution of the problem of 
normal traction given by these authors, when reduced to its simplest form, involves 
double integrals of simple harmonic functions of the coordinates. The integrals are of 
complicated form, and practically impossible to interpret, a fact which, without doubt, 
has had much to do with the neglect of the problem in later times, and the almost com- 
plete absence of attempts to establish the approximate theory on the basis of an exact 
solution. An even more serious defect of Lamm and Ciapryron’s solution is that the 
integrals, as they stand, do not converge. A flaw of this sort has often been treated 
lightly by physical writers, the non-convergence of an integral being regarded as due to 
the inclusion of an infinite but unimportant constant. In the present case, however, 
the infinite terms are not constant, but functions of the coordinates, and the modifica- 
tions necessary to secure convergence, so far from being unimportant, lead directly to 
the most significant terms of the solution. 
The next writer to deal with the exact problem was Sir W. Tuomson, who, at the end 
of the memoir in which he solved the problem of a spherical shell, indicated the form 
which the solution would take in the limiting case of a plate. His: method, if carried 
out, would lead to integrals of the same form as Lamers. 
Solutions of special problems have been given by other writers. Prof. Lamp has 
worked out the solution for an infinite solid subjected to normal pressure proportional 
to cos kx, and verified in this particular case some of the results of the approximate 
theory of thin plates (Proc. Lond. Math. Soc., vol. xxi., 1889-90). 
The history of the approximate theory is well known and th accessible, It will 
be sufficient here to refer to— 
(i) TopHunTER and Prarson’s History of the Elasticity and Strength of Materials. 
(ii) Ciexscn’s standard treatise, Théorie de Uélasticité des corps solides, as trans- 
lated by Satnt Venant ; in particular, Part I. chap. iii., and Sarnr Venanv’s brilliant note 
on § 73. 
(ii) Prof. Lovn’s Treatise on the Theory of Elasticity, 1892, — especially the 
historical introductions to both volumes. 
The various forms of the approximate theory rest partly upon the general equations 
of equilibrium, partly upon auxiliary hypotheses or physical principles, These 
principles are recognised as contained in the general equations, but on account of the 
TRANS. ROY. SOC, EDIN,, VOL, XLI. PART I, (NO. 8), 22 
